Analysis Seminar
Wednesday, 08/03/2023, 4 pm
Host: B. K. Das
Venue: Ramanujan Hall
Speaker: Haripada Sau
Affiliation: IISER Pune
Title: Multivariable Operator Theory: Rational Dilation,
Realization Formula, and Distinguished Varieties.
Abstract: In 1951, John von Neumann proved a pathbreaking inequality involving contractive linear transformations acting on Hilbert spaces and bounded analytic functions on the unit disk. The inequality -- now referred to as the von Neumann inequality -- has an extraordinary influence on the development of operator theory. A couple of years later, a simpler proof of the von Neumann inequality emerged when Sz.-Nagy proved his dilation theorem - the genesis of the dilation theory. Sz.-Nagy's discovery prompted several mathematicians to consider a multivariable generalization of the classical dilation theory. This is called the rational dilation problem -- the Holy Grail of spectral theory. In this talk, we shall discuss in brief the rational dilation problem corresponding to the unit disk, the bidisk, the symmetrized bidisk, and the tetrablock. We shall also discuss in brief a realization formula for bounded analytic functions on the disk, the bidisk, and the symmetrized bidisk. We shall see how a special class of algebraic varieties - the so-called distinguished varieties - becomes a natural object of study in the context of multivariable von Neumann inequality. A new characterization of such varieties will be presented. The last part of the talk will be about an ongoing project concerning a constrained two-variable dilation problem.